... But then, I'm unfamiliar with the term flattening used to describe the
process....

Nevertheless, the vowel in question here is, as has already been pointed out,
the small printed "a", a vowel completely familiar to me as a native of
Eastern New England. (Norwood MA to be specific, some 14 miles south and a
little west of Fenway Park.) I have it in half , laugh , path , as well as
in cart , barn , and the like. The vertical centered dot to the right of it
indicates a compensatory lengthening for the lack of postvocalic /r/. It
is included in all of the vowel charts I have handy from PEAS, the New
England and LAMSAS Handbooks and LAUM, and I would describe it as a lower low
front unrounded vowel. There is not much discussion of it in any of these
sources: Harold Allen says that it is found in cart and barn in eastern
New England. Incidentally, I also have the [a] vowel in father , but not in
pot and bother .

When I was at McDavid's NEH seminar in 1977 Mike Dressman told me that [au]
diphthongs were monophthongized as well as the [ai] diphthong, resulting in
[a:]. I'm still not comfortable with flattening.

In the early 1970s my brother and his wife moved from Norwood MA to Kingsport
TN.
They were suprised to heard the local cheerleaders and the crowd encouraging
the local football team to [fa:t], [fa:t], [fa:t].

I've been waiting for Don Lance to straighten us all out on this.

Thanks for the vote of confidence, David, but this one's pretty big -- a
melange of several things. I can make only a few comments, not "straighten
it all out" by any means. Flattening is a non-technical term that
Southerners use for their monophthingized [ai] -- [a]. As you point out,
the single dot indicates "half-lengthening," whatever the context or
dialect. In New England, "broad a" is the popular term used for your
vowels in half , cart , etc. The feature [+flat] has been used by some
phonologists, but at the moment I can't remember what the articulatory of
distributional characteristics of [+flat] are. But [+flat] and
"flattening" are certainly not the same thing. I don't have an explanation
for how "flat" and "broad" were chosen for these allophones/diaphones.

The people who monophthongize (or flatten, if you prefer) [au] to [a] are
not the same ones who [fa:t] hard on the football field. I hear /au/
"flattening" in Midwestern dialects when the /au/ diphthong precedes an
unstressed syllable, as in "cloddy" for "cloudy" and "consul" for "council."

I was gonna continue just lurking on this one, till David Hwaet prodded me.
Is this what you wanted?